Definition:Uniformly Continuous Semigroup

Definition

Let $\GF \in \set {\R, \C}$.

Let $X$ be a Banach space over $\GF$.

Let $\family {\map T t}_{t \ge 0}$ be a $\hointr 0 \infty$-indexed family of bounded linear transformations $\map T t : X \to X$.

Let $\struct {\map B X, \norm {\, \cdot \,}_{\map B X} }$ be the space of bounded linear transformations equipped with the canonical norm.

We say that $\family {\map T t}_{t \ge 0}$ is uniformly continuous if and only if:

$\ds \lim_{t \mathop \to 0^+} \norm {\map T t - I}_{\map B X} = 0$

Also see

• Semigroup of Bounded Linear Operators Uniformly Continuous iff Continuous as Map from Non-Negative Reals to Bounded Linear Operators

• Results about uniformly continuous semigroups can be found here.

Sources

• 1983: Amnon Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations ... (previous) ... (next): $1.1$: Uniformly Continuous Semigroups of Bounded Linear Operators